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\input amssym.def            % small letters for UNIX,  not: AMSsym.def

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum


\nopagenumbers

\vglue 10pt


\cl{\bf Parabola}

\noindent
See also: Ellipse, Hyperbola, Conic Section and their ATOs, and 
in the Category Surfaces see:
Conic Sections and Dandelin Spheres

\lf
The parametric equations for the Parabola are \lf
$ x(t) := t^2/4p    \lf
  y(t) := t $,                 \lf
where $p = aa/4,$
\lf
so the Parabola visualizes the graphs of the two functions
$ y(x) := \sqrt{4p\cdot x}$ and $ x(y) := y^2/4p$.
\lf
The vertical line $x=-p$ is called the directrix and the point
$(x,y)=(p,0)$ is called focal point of the Parabola. The distance from a
point $(x,y= \sqrt{4p\cdot x})$ on the Parabola to the directrix is $(x+p)$,
and this is the same distance as from $(x,y= \sqrt{4p\cdot x})$
to the focal point $(p,0)$, because $(x-p)^2 + y^2 = (x+p)^2$.
\lf
The point $(p,0)$ is called ``focal point'', because light rays which
come in parallel to the x-axis are reflected off the Parabola so that
they continue to the focal point. This fact is illustrated in the program.
It gives the following ruler construction of the Parabola:
\lf
Prepare the construction by drawing x-axis, y-axis, directrix and focal
point F.
Then draw any line parallel to the x-axis and intersect it with the directrix
in a point   S. The line orthogonal to the connection SF and through its
midpoint
is the tangent of the Parabola and intersects therefore the incoming ray
in the point of the Parabola which we wanted to find.

\lf
The same construction works for Ellipse and Hyperbola, if the
directrix is replaced by a circle of radius 2*a around one focal point. The
curve is the set of points which have the same distance from this circle
and the other focal point. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Text Nr 12 ends here %%%%%%%%%%%%%%%%
\LF
The Action Menu of the Parabola has an entry ``Show Normals Through Mouse Point''.
This illustrates an unexpected property of the Parabola. One may already
be surprised that at the intersection points of normals always {\bf three} normals
meet. We know no other curve which is accompanied by such a net of normals. The
surprise should increase if one looks at the $y$-coordinates of the parabola
points from where three such intersecting normals originate: these $y$-coordinates
add up to $0$! In other words, the intersection behaviour of the normals reflects
the addition on the $y$-axis. 

\lf
The explanation of where this intersection property comes from is quite 
interesting. The normals of the Parabola are the tangents to its evolute,
the semi-cubical parabola, a singular cubic curve (see Cuspidal Cubic). 
So the intersection property of the parabola normals can be thought of
as defining an addition law for the evolute, and as such it
is a simpler limiting case of the addition law that exists on any cubic curve.



\bye